In figure, if AB ‖ DC and AC, PQ intersect each other at the point 0. Prove that 0A. CQ = OC.AP.

Question

Philoid · Accepted Answer

Given,

AC and PQ intersect each other at the point O and AB‖DC.

in ∆AOP and ∆COQ,

∠AOP = ∠COQ [vertically opposite angles]

∠APO = ∠CQO [since, AB‖DC and PQ is transversal, so alternate angles]

∴ ∆AOP ∼ ∆COQ [by AAA similarity criterion]

Then, OA/OC = AP/CQ

[since, corresponding sides are proportional]

⇒ OA × CQ = OC × AP

Hence Proved.

In figure, if AB‖DC and AC, PQ intersect each other at the point 0. Prove that 0A. CQ = OC.AP.